The purpose of these notes is to describe the notion of an Euler system, a collection of compatible cohomology classes arising from a tower of fields that can be used to bound the size of Selmer groups. There are applications to the study of the ideal class group, Iwasawa’s main conjecture, Mordell-Weil group of an elliptic curve, X (the Safarevich-Tate group), Birch-Swinnerton-Dyer conjecture,...

We present a Mordell-Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. study $X_0(N)(\mathbb{Q}(\sqrt{d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and $\lvert d \rvert < 100$.

Let Ep be an elliptic curve over a prime finite field Fp, p ≥ 5, and Pp, Qp ∈ Ep(Fp). The elliptic curve discrete logarithm problem, ECDLP, on Ep is to find mp ∈ Fp such that Qp = mpPp if Qp ∈ 〈Pp〉. We propose an algorithm to attack the ECDLP relying on a Hasse principle detecting linear dependence in Mordell-Weil groups of elliptic curves via a finite number of reductions.

We explicitly define the set of algebraic points a hyperelliptic curve $\mathcal{C}_q$ any given degree having affine equation $y^2=(x-2 q)\left(x^2-2 q^2\right)\left(x^2+2^2 q^2\right)$. Such is described in [4], where it shown that Mordell-Weil group finite for $q \equiv 13$ [24]. Furthermore, generators torsion these curves are explained. Received: March 25, 2023Accepted: July 3, 2023

In his work on Diophantine equations of the form y2=ax4+bx3+cx2+dx+e, Fermat introduced the notion of primitive solutions. In this expository note we intend to interpret this notion more geometrically, and explain what it means in terms of the arithmetic of elliptic curves. The specific equation y2 =x4 + 4x3 + 102 +20x+ 1 was used extensively by Fermat as an example. We illustrate the nowadays ...

The main result of this note is that there are at most seven rational points (including the one at infinity) on the curve CA with the affine equation y = x+A (where A is a tenth power free integer) when the Mordell-Weil rank of the Jacobian of CA is one. This bound is attained for A = 18 .

Mordell-Weil groups of different abelian fibrations of a hyperkähler manifold may have non-trivial relation even among elements of infinite order, but have essentially no relation, as its birational transformation. Precise definition of the terms ”essentially no relation” will be given in Introduction.